Finally 3 follows from the fact that the functors \(N\)  and \(T\)  preserve
products and the well known fact that \(G//G\)  is a group object in \(\mathit {Gpd}({E})\) .

Specialized to the case when \({C}= \mathit {Set}\) , the principal twisted cartesian product \(T N(G//G)\rightarrow T NG\) 
constructed in Lemma REF  reduces to the
classical principal twisted cartesian product \(WG\rightarrow \overline{W}G\)  as described for instance in
Corollary 21.8 of [1]}. Recall that \(\overline{W}G\)  is the
simplicial set with \(\overline{W}G_ 0 = 1\)  and
\(\overline{W}G_n = G_{n-1}\times G_{n-2}\times \cdots \times G_0\) 
